I know that
[tex]z\frac{d}{dz}K_v(z)+vK_v(z)=-zK_{v-1}(z)[/tex]
but I was confused when we have more complicated arguments such as
[tex]z=\sqrt{x^2+x}[/tex].
But after your posting, I have now a simple method to move from simple to more complicated arguments. So, I can say the following:

I know that
[tex]z\frac{d}{dz}K_v(z)+vK_v(z)=-zK_{v-1}(z)[/tex]
but I was confused when we have more complicated arguments such as
[tex]z=\sqrt{x^2+x}[/tex].
But after your posting, I have now a simple method to move from simple to more complicated arguments. So, I can say the following: